# What is, really, the stable homotopy category?

When you try to understand the fuss behind the new good categories of spectra that arose on the 90's, you read things such as the following paragraph written by Peter May (from "The Hare and the Tortoise"):

All consumers are now in agreement: Mike's stable homotopy category is definitively the right one, up to equivalence. However, the really fanatical hare demands a good category even before passage to homotopy, with all of the modern bells and whistles. The ideal category of spectra should be a complete and cocomplete Quillen model category, tensored and cotensored over the category of based spaces (or simplicial sets), and closed symmetric monoidal under the smash product. Its homotopy category (obtained by inverting the weak equivalences) should be equivalent to Mike's original stable homotopy category.

Here "Mike" is Michael Boardman. Question number zero would be, could anybody share Boardman's "Stable homotopy theory" mimeographed notes where he introduces said category? Both that and Vogt's "Boardman's stable homotopy category" are hard to find, so it's hard for me to know what they are talking about.

But suppose I know what Boardman's stable homotopy category is. Why should I agree that this is "the" right stable homotopy category, up to equivalence?

I am guessing that the right way to formalize what I mean is: elabore a desiderata for a Stable Homotopy Category, prove that Boardman's satisfies them, and then prove that any two categories satisfying those axioms are equivalent.

Has that been done? If so, where?

I can try to answer my question. Margolis' book "Spectra and the Steenrod Algebra" from 1983 does have such a list of axioms, in section 1.2.

Is that an idiosyncratic list of axioms or is it really what homotopy theorists of the time would agree that it's exactly what they would have wanted?

I know this is perhaps argumentative. But the following is not. At the end of said chapter in Margolis' book, he conjectures that any two categories satisfying those axioms are equivalent. But at the time, it apparently wasn't established.

Has it been established since?

But maybe there is another characterization of "the" stable homotopy category, of which Boardman's (or Adams', or...) would be an example; I'd be interested by that, too.

I am mildly aware of the fact that there is a stable $\infty$-categorical universal property. That is certainly interesting, but I would be interested to see how it could be formulated in older language (model categories?) since such a formulation is, in the spirit of my question, anachronistic. (Not that I would find it uninteresting).

• For the last question, see mathoverflow.net/questions/67227/… Jan 11, 2016 at 13:24
• The third part of Adams's book "Stable homotopy and generalized homology" is a standard source for Boardman's category of spectra. Jan 11, 2016 at 14:36
• You might also want to look at Hovey-Palmieri-Strickland's Axiomatic Stable Homotopy monograph. It continued what Margolis started and wrote down definitive axioms, then developed the theory from the axioms and provided a number of examples, including the stable homotopy category and the derived category of a ring. Jan 11, 2016 at 21:12
• Chapter V from Boardman's manuscript is available here: math.ucr.edu/~res/inprogress/Boardman-V.pdf Jan 12, 2016 at 10:42
• @Bruno: May's quote "[Frank Adams'] category had only connective spectra in it" refers to the Springer Lecture notes (no. 3) version from the 1961 Berkeley lectures, not the blue book version from the 1971 Chicago lectures. Mar 10, 2021 at 14:40

I am mildly aware of the fact that there is a stable ∞-categorical universal property. That is certainly interesting, but I would be interested to see how it could be formulated in older language (model categories?)

Symmetric spectra have a model categorical universal property: they form the initial stable monoidal model category (theorem of Shipley). A subtlety is that this statement holds with the positive stable model structure, but at least this is Quillen equivalent to the usual one and so captures the same homotopy theory.

At the end of said chapter in Margolis' book, he conjectures that any two categories satisfying those axioms are equivalent. But at the time, it apparently wasn't established.

Has it been established since?

Shipley also shows that any category satisfying Margolis's axioms that actually comes from a stable monoidal model category, must be equivalent to the stable homotopy category (as a monoidal category). That pretty much resolves Margolis's conjecture I'd say, since by now everyone agrees that triangulated categories are just not enough for these kinds of questions.

The relevant paper of Shipley is called Monoidal Uniquness of Stable Homotopy theory.

Here are some funky bonus facts:

• Given an object in a symmetric monoidal $\infty$-category, there exists a universal functor which is initial among symmetric monoidal functors which invert the given object. The symmetric monoidal $\infty$-category of spectra is obtained from pointed spaces with smash product by formally inverting $S^1$ in this way. (The symmetric monoidal $\infty$-category of pointed spaces is in turn obtained from spaces with cartesian product by freely pointing.) See this paper of Robalo.

• Given a Grothendieck derivator (another way to keep track of homotopy (co)limits in a homotopy category), there exists a universal stable derivator associated to it. Taking the derivator of spaces, this gives the stable derivator of spectra. Also, every stable derivator is canonically enriched over the derivator of spectra. See the appendix of this paper of Cisinski and Tabuada. The language of derivators is very elementary (no homotopy theory used!) and is pretty old (early 80's).